ISE/OR 762 Stochastic Simulation Techniques

Transcription

1 ISE/OR 762 Stochastic Simulation Techniques Topic 0: Introduction to Discrete Event Simulation Yunan Liu Department of Industrial and Systems Engineering NC State University January 9, 2018 Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

2 Why Simulation? Simulation is used to simulate the operations of various kinds of real-world systems or processes with the aid of a computer. We make assumptions about how a system works; these assumptions constitute a model. Models are used to gain some understanding of how a system behaves. Analysis approach based on system complexity: Simple models: use analytical (mathematical) methods to obtain exact information on questions of interest. Complex models: use simulation to evaluate the model numerically. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

3 System Analysis System: a collection of entities, e.g. people or machines, that act and interact together toward the accomplishment of some logical end. State: collection of variables necessary to describe a system at a particular time, relative to the objectives of a study. A system can be discrete or continuous depending on how the state variables change with respect to time. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

4 Application of Simulation Designing and analyzing manufacturing and inventory systems Determining hardware requirements or protocols for communication networks Determining hardware and software requirements for computer systems Designing and operating transportation systems such as airports, freeways, sports and subways Evaluating designs for service organizations such as call centers, fast-food restaurants, hospitals and post offices Analyzing financial or economic systems; pricing complex financial instruments Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

5 Different Kinds of Simulation Static v.s. dynamic Does time have a role in the model? Continuous-change v.s. discrete-change Can state change continuously, or only a discrete points in time? Deterministic v.s. stochastic Is everything for sure or is there uncertainty? By hand v.s. by computer Is the model complicated so we need a computer? Most operational models: Dynamic, discrete-change, stochastic, computer-based Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

6 Discrete Event Simulation Objective: Model a system as it evolves over time by a representation in which the state variables change instantaneously at separate points in time, e.g. a queue. Characteristics: Keep track of the simulated (random) time as simulation proceeds. Advance simulated time via a simulation clock. Simulation clock can be advanced either when an event occurs or at a fixed increment of time. Sample system states as time evolves to prepare for output estimators. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

7 Example: A Single-Server Queue Random elements: Ai = time of arrival of the i th customer; ti = A i A i 1 = interarrival time (t 0 = 0); Vi = processing (service) time of i th customer; Wi = waiting time of i th customer (excluding service); Di = A i + W i + V i = departure time of i th customer. Possible events: arrivals and departures. Advance simulation clock when an event occurs ei = time of the i th event. V i s and t i s distributed according to a known distribution. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

8 Estimating Useful Performance Metrics Estimate the average waiting time of customers N(t) = number of customers that have arrived in [0, t] Ŵ (T ) = 1 N(T ) N(T ) i=1 Estimate the average number of customers in the system W i Q(t) = total number of customers in the system at time t Q(T ) = 1 T T 0 Q(t)dt. Estimate the average utilization ρ(t ) = 1 T T 0 B(t)dt, B(t) = 1 {Q(t)>0}. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

9 An Example with Numbers Set T = 10 Initialize system so that there are no customers at time 0 Generate arrivals and service times according to their distributions Arrival times: 1.5, 2.4, 3, 3.6, 6, 7.7, 9.2, 11 Processing times: 1, 1.5, 1, 2, 0.5, 3.5, 1.5, 1 Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

10 Simulation Outputs Compute desired performance functions: N(T ) = 7 Ŵ (T ) = = Q(T ) = 1 ( )+2 ( )+3.4 ρ(t ) = = 1.26 [ ] = 0.83 Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

11 Monte Carlo Simulation Definition: Computational algorithms that rely on repeated random sampling to obtain numerical results. (A simulation estimator based on strong law of large numbers.) Named after the gambling hot spot in Monaco: chance and random outcomes are central to the modeling technique, much as they are to games like roulette, dice and slot machines. First developed by Stanislaw Ulam and John Von Neumann. Applications: Evaluate complicated integrals (multiple dimensions), pricing complex financial instruments (options) Example: Estimate the value of I b a g(x)dx. Generate X Unif(a, b), set Y = (b a)g(x ). Then 1 n n Y i E[Y ] = (b a)e[g(x )] = (b a) i=1 b a g(x) 1 b a dx = b a g(x)dx. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

12 Buffon s Needle Problem: Precomputer Simulation in 1777 Buffon is remembered in the history of probability theory for his famous needle problem - the first example of a simulation experiment Buffon s Needle Problem: If a floor has equally spaced parallel lines a distance d apart and if a needle of length l is tossed at random on the floor where l d, then what is the probability that the needle will intersect a line? Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

13 Buffon s Needle Problem: Precomputer Simulation in 1777 The distance from the head of a needle to a line is X Unif(0, d). Conditioning on Θ: P (Needle intersects a line Θ = θ) = P (X [0, l sin θ]) = Unconditioning on Θ yields: P (Needle intersects a line) = = π 2 0 π 2 0 l sin θ 0 1 d P (Needle intersects a line Θ = θ) sin θ l d When l = d, P (Needle intersects a line) = 2 π = l dθ = π πd = sin θ l dx =. d 1 π/2 dθ Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

14 Buffon s Needle Problem: Precomputer Simulation in 1777 Toss a needle for a large number N times. Define independent r.v. s: { 1, if i X i = 1 {i th needle crosses a line} = th needle crosses a line; 0, otherwise. According to the strong law of large number (SLLN): 1 lim N N N X i = 2 π i=1 so that π 2 1 N N i=1 X i for N large. Remarks: Buffon s needle-tossing experiment can be conducted by hand. It is the earliest example of using independent replications of a simulation to approximate an important physical constant. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

15 Simulations in TV Shows British Sci-Fi TV drama on Netflix: Black Mirror Season 4, Episode 4: Hang the DJ An algorithm simulating the dating outcomes of 1000 independent digital copies of the consciousness of Frank and Amy. The app promises a 99.8% percent perfect match. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

16 Issues Arising in Monte Carlo Simulation How to generate needed input random objects? How many computer experiments do we need? How to compute expectations associated with limit stationary distributions? How to exploit problem structure to speed up the computation? How to efficiently compute probabilities of rare events? How to estimate sensitivity of a stochastic model to changes in a parameter? How to use simulation to optimize our choice of decision parameters? Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

17 Advantages and Disadvantages of Simulation Advantages: Simulation allows to estimate the performance of complex, real-world, systems that cannot be evaluated analytically. Allows us to study long time frames in compressed time. Provides better control of experimental conditions. Disadvantages: Lacking of insights. Expensive and time-consuming to develop. Produces only estimates (not exact) of a model s true characteristics. Implementation errors; Sampling errors; Machine errors; Statistical errors. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

18 Pitfalls of Simulation Lack of well defined objectives. Failure to collect good system data. Failure to account correctly for sources of randomness. Use of arbitrary distributions. Assume independence when it is not present. Not running enough simulations. Treating simulation results as true answers. Replacing random variables by their means. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

19 Danger of Replacing Random Variables by Their Means Example: Consider a manufacturing system consisting of a single machine. Raw parts arrive according to a Poisson process with rate λ = 1 per min. Processing times are exponentially distributed with mean 1/µ = 0.99 min. This is an M/M/1 queue with traffic intensity ρ = 0.99 (ISE760) The long-run average number of customers in the system 1 E[Q( )] = lim t t The long-run average waiting time 1 E[W ] = lim n n t 0 Q(u)du = ρ 1 ρ = 99. n W i = 1 ρ = minutes. µ 1 ρ i=1 However, if we replace both distributions by their corresponding means, then The average number of customers in system 1; No part is ever delayed in the queue! Conclusion: Ignoring randomness may result in extremely inaccurate estimates. Yunan Liu (NC State University) Stochastic Simulations January 9, / 20

20 Rest of the Class... Review of probability theory (Topic 1). How to generate the random elements (variable, process, permutation, etc.) following various distributions (Topics 2 3). How to use Monte Carlo simulation to price path-dependent financial instruments (Topic 4). How to use the generated random elements to characterize the dynamics of a realistic system (Topic 5). Given a realization of a system, how to design effective procedures to estimate desired performance measures (Topic 6). When many sample paths (realizations) are needed, how to design efficient method to reduce number of samples (shortening the total computation time, Topic 7). How to select appropriate probability distributions (and their parameters) for the simulation algorithms (Topic 8). Yunan Liu (NC State University) Stochastic Simulations January 9, / 20